MACHINE. 



a weif'ht tliat is double of W couM be fuflained bv the 

 power P. 



l-'or another example ; fuppofe a fluid, moving with the 

 velocity and direction A C, [Jig- ,>•) ftrike the plane 

 C E ; and fuppofe that this plane moves parallel to itfclf 

 in the direction C B, perpendicular to C A, or that it 

 cannot move in any other direction. Tiien let it be re- 

 quired to find the mod advantageous pofition of the plane 

 C E, that it may receive the greatefl impulfe from the 

 aftion of the fluid. Let A P be perpendicular to C E in 

 P, draw A K pa-allel to^ B, and let P K be perpendicu- 

 lar upon it in K, and A K -A-ill mcafure the force with "which 

 any particle of the fluid impels the plane E C, in the direc- 

 tion C B. For the force of any fuch particle being re- 

 prefented by A C, let this force be refolved into A Q, 

 parallel to E C, and A P perpendic ilar to it ; and it is 

 manifeft, that tlie latter A P only has any effeft upon the 

 plane C E. Let this force A P be refolved into tlie 

 force A L perpendicular to C B, and the force A K pa- 

 rallel to it ; then it is manifeft, that the former, A L, 

 has no effect in prom.oting the motion of the plane in the 

 direction C B ; fo that the latter A K, only, meafures the 

 effort hy which the particle promotes the motion of the 

 plane C E in the direction C B. Let E M and E N be 

 perpendicular to C A and C B, in M and N ; and the 

 number of particles, moving with direftions parallel to 

 A C, incident upon the plane C E, v.-ill be as E M. 

 Therefore the effort of the fluid upon C E being as the 

 force of each particle, and the number of particles to- 

 gether, it will be as A K X E M ; or, becaufe A K 



M F= y FN 

 is to A P (= E M) as E N to C E, as cE"^' 



fo that C E being given, the problem is reduced to this, to 

 find wiien E M' x E N is the greateft poffible, or a maxi- 

 mum. Bntbecaufe the fumtif E M-and of E N' (= C M') 

 is given, being always equal to C E', it follows that E N' 

 X E M- is greateft when E N" = j C E ; in the fame 

 manner as it was demonllrated above, that when the fum of 

 AC and CB (;%• I-) was given, AC x C B' was 

 greateft, when AC = i A B. But when E N' X E M' is 

 greateft, its fquare root EN x E M' isof neceflity at the 

 fame time greateft. Tiierefore the aftion of the fluid upon 

 the plane C E, in the direflion C Bj is greateft when E N" 

 = i C ES and confequently E M '=; * C E ; ihat is, when 

 E M, the fine of the angle ACE, in which the ft;ream 

 ftrikes the plane, is to the radius, as the ./ 2 to ,/ 3 ; in 

 which cafe it eaiiiy appears, from the trigonometrical tables, 

 that this angle is of j'4° 44'. 



Several ufeful problems in mechanics may be refolved by 

 ■what was ftiewn in the preceding paragraph. If we re- 

 prefent the velocity of the wind by A C, a feftion of the 

 fail of a windmill, perpendicular to its length by C E, 

 as it follows from the nati>re of the engine, that its axis 

 ought to be turned dircftly towards the wind, and the 

 fail can only move in a direftion perpendicular to the axis, 

 it appears, that when the motion begins, the wind will have 

 the greattlt effect to produce this motion, when the 

 angle A C E, in which the wind ftrikes the fail, is of 

 54 44'. Ib the fame manner, if C B reprefent the di- 

 rection of the motion of a fnip, or the pofition of her keel, 

 abftracting from her lee-way, aad A C be the direftion of 

 the wind, perpendicular to her way, then the moil ad- 

 vantageous petition of the fail C E, to promote her motion 

 in the direftion C B, is when the angle ACE, in which 

 the wind ftrikes the fail, is of i^^' 44'. The belt pofition of 

 the ladder, where it may have the greatell effeft in turning. 



round the ftiip, is determined in like tnanner, and the fame 

 angle enters likewife into the determination of the figure of 

 the rhombufes tjiat form the bafes of the cells in which 

 the bees depofit their honey in the moll frugal manner. 

 (See HoxEY-Comi ) But it is to be carefully obferved, that 

 when the fine of the angle A C E is to the radius as ^/ 2 to 

 ^/ 3 ; or, which is the fame thing, when its tangent is to 

 the radius as the diagonal of afquarc to its fide ; this is the 

 moft advantageous angle only at the beginning of the 

 motion of the engine ; fo that the fails of a common 

 windmill osight to be fo iiiuated, that the wind may in- 

 deed ftrike tliem in a greater angle than that of 54' 44'. 

 For it IS demonftrablc, that when any part of the engine 

 has acquired the velocity c, the effort of the wind upon 

 that part will be greatell, when the tangent of the angle 

 in which the wind ftrikes it, is to the radius, not as the 



V 2 to I, but ^/ 2 x -- — X — to I, the velocity of the 



4 a' 2 a 



wind being reprefented by a. If, for example, c ■= iaf 

 then the tangent of the angle ACE ought to be double 

 of the radius ; that is, the angle ACE ought to be of 

 63^26'. If c = a; then ACE ought to be of 74^ 19'^ 

 This obfervation is of the more importance, becaufe, in 

 this engine, the velocity of the parts of the fail remote 

 from the axis bears a confiderable proportion to the ve- 

 locity of the wind, and perhaps fometimes is equal to- 

 it ; and becaufe a learned author, Daniel Bernouilli, has 

 drawn an oppofite conclufion from his computations in 

 his hydrodynamics, by miftaking a minimum for a maxi- 

 mum ; where he infers, that the angle in which the wind 

 ftrikes the fail, ought to decreafe as the diftance from 



the axis of motion increafes 

 ought to ftrike 



thal^ if ;: = a, the wind 

 in an angle of 4^'' ; and that if the fail 

 be in one p'ane, it ought to be inclined to the wind, at 

 a medium, in an angle of 50". How he fell into thefe 

 mi'lakcs, is fhewn by Maclaurin, in his Fluxions, 5 914. 



In like mann.r, though the angle -\ C E of 54 44' 

 be the moit advantageous at the beginning of the motion, 

 when a ihip fails with a fide wind ; yet it ought to be 

 enlarged afiervvards as the motion increafes. In general, 

 let A a (Jig. 3 ) parallel to C B, be to A C, as the ve- 

 locity whicti tire engine has already acquired in the direc- 

 tion C B, to that of the ftream ; upon A C produced,, 

 take A D to A C as 4 to 5, draw D G parallel to C B, 

 and let a circle delcribed from the centre C with the ra- 

 dius C a, meet D G in ^ ; and the plane C E fhall be in 

 the moft advantageous fituation for promoting the motion, 

 of die engine, when it bifefts the angle a C g. 



It is generally fuppofed, that a direft wind always pro- 

 motes the motion of a fliip, the fail being perpendicular to 

 the wind, more than any fide-wir.d ; and this has been af- 

 firmed in feveral late ingenious treatifes ; but, to prevent 

 miftakes, we are obliged to obfcri-e, that Maclaurin has 

 demon ftrated the contrary in his Trealife of Fluxions^ 

 J 919 ; where other inftances of this fecond general problem 

 in mechanics are given, to which we refer. See Maclau- 

 rin's Account of fir Ifaac Newton's Philofophical Difcove- 

 ries, book ii. chap. 3. p. 1 73. 



Let <? denote the abfolute effort of any movijig force>, 

 when it has no velocity, and fuppofe it not capable o£ 

 any effort when the velocity is W ; let F be the effort an-.- 

 fwering to the velocity V, then if the force be uniform, w&; 

 ftiali have 



